Kapil is programming a robot to always know its distance from its charging base by following these steps: $ \text{Step }1)$ Save $b$, which is the current distance from the base. $ \text{Step 2})$ Face charging base and turn $\theta^\circ$ to the right. $ \text{Step 3)}$ Move $x$. $ \text{Step 4)}$ Compute new distance from the base. $ \text{Step 5)}$ Go back to step $1$. For example, it might happen that when the robot gets to step $4$ in its program, $ b = 70$ units, $ \theta = 60^\circ$, and $ x = 50$ units. In the example above, what would the robot's new distance from its base be? Do not round during your calculations. Round your final answer to the nearest unit.
Converting the problem into geometrical terms Our problem can be modeled by the following triangle $\triangle ABC$, where we want to find $AC=d$. $60^\circ$ $50$ $70$ $d$ $A$ $B$ $C$ Since we are given two side lengths and the angle measure between them, we can use the law of cosines. Using the law of cosines $\begin{aligned} (AC)^2&=(AB)^2+(BC)^2-2AB\!\cdot\! BC\!\cdot\!\cos(B)\\\\ d^2&=50^2+70^2-2\cdot 50\cdot 70\cdot\cos(60^\circ) \gray{\text{Substitute}}\\\\ d&=\sqrt{50^2+70^2-2\cdot 50\cdot 70\cdot\cos(60^\circ)}\\\\ d&\approx 62 \end{aligned}$ The answer The robot's new distance from its base would be $62$ units.